# Exponents

### Topics

Exponents are used in most mathematics. They occur in polynomials, where the largest exponent defines the order of the polynomial. For example:

y = 3x5 - 7x2 + 2x + 5 is a fifth order polynomial, because 5 is the largest exponent.

In some cases, the variable is located in the exponent. These are called exponential equations. For example:

y = 5x - 4 is an exponential equation because x is in the exponent.

Exponential equations are used in finance to find growth of an investment and the cost of a loan with compound interest. Anthropologists use exponential equations to do Carbon-14 age dating. Radiologist in hospitals use exponential equations to determine radioactive decay of materials used for cancer treatment. Biologists use exponential equations to predict population growths for organisms from humans to bacteria.

## Definition of Exponents

Exponents are used to show repeated multiplication in an equation.  For example:

a • a • a • a • a = a5

If a is a real number and n is a positive integer, then

an = a • a • a • ... • a  (n factors)

where n is the exponent and a is the base. The expression an is read "a to the nth power".

Exponential equations are the inverse of logarithms. The properties are very similar for both exponents and logarithms.

## Properties of Exponents

Let a and b be real numbers, variables, or algebraic expressions and let m and n be integers. All denominators and bases must be nonzero.

Property

Example

1.  aman = am + n 3234 = 32 + 4 = 36 = 729
 2. am = am - n an
 y7 = y7 - 4 = y3 y4
 3. a-n = 1 an
 x-3 = 1 x3
4.  a0 = 1,   a 0 (x3 - 4)0 = 1
5.  (ab)n = anbn (4x)3 = 43x3 = 64 x3
6.  (am)n = amn (y3)4 = y3(4) = y12
 7. ( a ) m = am b bm
 ( 2 ) 4 = 24 = 16 3 34 81
8.  |a2| = |a|2 = a2  |(-3)2| = |-3|2 = 32 = 9

Comments, questions, or you just need some help? Send an e-mail to Mrs. Shearer Last Updated: 10/15/2010